Stochastic processes are mathematical objects that offer a probabilistic representation of
how some quantities evolve in time. In this thesis we focus on estimating the trajectory and
parameters of dynamical systems in cases where only indirect observations of the driving
stochastic process are available. We have first explored means to use weekly recorded
numbers of cases of Influenza to capture how the frequency and nature of contacts made
with infected individuals evolved in time. The latter was modelled with diffusions and
can be used to quantify the impact of varying drivers of epidemics as holidays, climate,
or prevention interventions. Following this idea, we have estimated how the frequency of
condom use has evolved during the intervention of the Gates Foundation against HIV in
India. In this setting, the available estimates of the proportion of individuals infected with
HIV were not only indirect but also very scarce observations, leading to specific difficulties. At last, we developed a methodology for fractional Brownian motions (fBM), here a
fractional stochastic volatility model, indirectly observed through market prices.
The intractability of the likelihood function, requiring augmentation of the parameter
space with the diffusion path, is ubiquitous in this thesis. We aimed for inference methods
robust to refinements in time discretisations, made necessary to enforce accuracy of Euler
schemes. The particle Marginal Metropolis Hastings (PMMH) algorithm exhibits this mesh
free property. We propose the use of fast approximate filters as a pre-exploration tool to
estimate the shape of the target density, for a quicker and more robust adaptation phase
of the asymptotically exact algorithm. The fBM problem could not be treated with the
PMMH, which required an alternative methodology based on reparameterisation and advanced Hamiltonian Monte Carlo techniques on the diffusion pathspace, that would also
be applicable in the Markovian setting
